# Evaluate the following integrals. \int \frac{e^{x}}{4e^{x}+6}dx

Evaluate the following integrals.
$\int \frac{{e}^{x}}{4{e}^{x}+6}dx$
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veiga34

Step 1
To evaluate: $\int \frac{{e}^{x}}{4{e}^{x}+6}dx$
Solution:
Let substitute $t=4{e}^{x}+6$
Differentiating both sides,
$\frac{dt}{dx}=\frac{d}{dx}\left(4{e}^{x}+6\right)$
$\frac{dt}{dx}=4{e}^{x}+0$
$\frac{dt}{dx}=4{e}^{x}$
$\frac{dt}{4{e}^{x}}=dx$
Step 2
Substituting the values in given integral,
$\int \frac{{e}^{x}}{4{e}^{x}+6}dx=\int \frac{{e}^{x}}{t}\cdot \frac{dt}{4{e}^{x}}$
$=\frac{1}{4}\int \frac{1}{t}dt$
$=\frac{1}{4}\mathrm{ln}|t|+C$
Put back $t=4{e}^{x}+6$
$\int \frac{{e}^{x}}{4{e}^{x}+6}dx=\frac{1}{4}\mathrm{ln}|4{e}^{x}+6|+C$
Hence, required answer is $\frac{1}{4}\mathrm{ln}|4{e}^{x}+6|+C$.

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Mary Herrera

$\int \frac{{e}^{x}}{4{e}^{x}+6}dx$
We put the expression $\mathrm{exp}\left(x\right)$ under the differential sign, i.e.:
${e}^{x}dx=d\left({e}^{x}\right),t={e}^{x}$
Then the original integral can be written as follows:
$\int \frac{1}{4t+6}dt$
$\int \frac{1}{4x+6}dx$
Calculate the table integral:
$\frac{1}{2}\int \frac{1}{2x+3}dx=\frac{\mathrm{ln}\left(2x+3\right)}{4}$
$\mathrm{ln}\left({\left(2x+3\right)}^{\frac{1}{4}}\right)+C$
To write down the final answer, it remains to substitute $\mathrm{exp}\left(x\right)$ instead of t.
$\frac{\mathrm{ln}\left(2{e}^{x}+3\right)}{4}+C$

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